Introduction
The electrical conductivity of geological materials
varies over many orders of magnitude, and most materials have distinctive
ranges of conductivity. Electromagnetic
induction (EMI) is an effective technique for measuring electrical properties
but, in general, is complex in operation and interpretation.
Compact devices greatly simplify EMI, in that they
are most portable and operate typically at low induction number (LIN). With relative ease, LIN data yield reliable
interpretations of the conductivity and spatial extent of geological materials.
Electrical
Conductivity
The siemen, S, is the SI unit of conductance. Also known as the mho (i.e. inverse of ohm),
the siemen is defined as A/V, where A is the current in amperes that flows in
an electric circuit when a voltage V, in volts, is applied.
An electrical circuit in the earth has a
cross-sectional area through which the current flows, and a length over which
the voltage is applied. For a measure
that is independent of circuit dimensions, we divide current by area and
voltage by length (i.e. A/m2 / V/m) and obtain S/m, which is the SI
unit of conductivity.
The following chart (after Palacky, 1987) shows that
the conductivities of geological materials span many orders of magnitude, and
that a convenient unit for terrain conductivity is the millisiemen per metre,
or mS/m.
Figure 1: Conductivities of
Geological Materials.
The upper portion of the chart shows conductivities
of the various forms of water and of frozen soil, the conductivity of which is
governed by the ice it contains.
Unconsolidated materials, both transported and residual, are found in
the central level of the chart, below which are lithified materials.
Geological materials that are moist typically
conduct current through the movement of ions.
The ions may be in solution, or associated with surficial charges of
some clays, etc. The conductivity of
moist material shows linear proportionality to its temperature. The seasonal effect of temperature
fluctuations in a moderately conductive material is typically several
mS/m. Freezing a material tends to drop
its conductivity exponentially, usually by at least one order of magnitude.
Native metals, sulfides and graphite conduct current
through the movement of electrons. With
such materials, conductivity shows a small inverse proportionality to
temperature.
Electromagnetic
Induction
EMI instruments transmit a primary magnetic field,
which induces electrical current in the earth.
The current in the earth generates a secondary magnetic field, which is
sensed by the receiver of the instrument.
The characteristics of the secondary field indicate the conductivity of
the earth.
EMI techniques are distinguished by the nature of
the primary field. Transient (or
time-domain) EMI uses an intermittently pulsed primary field, and measures the
change in the secondary field at various times between pulses. Continuous (frequency-domain and LIN) EMI
uses a continuously varying primary field, and measures the amplitude and phase
of the secondary field relative to the primary field.
Most continuous EMI instruments transmit sinusoidal
primary fields at a fixed frequency, as depicted in the following figure. Such transmission is efficient, which
facilitates the acquisition of accurate data with instrumentation that is
relatively light and compact. The
figure also shows a secondary field that represents the response of an earth
that contains conductive material.
Figure 2: Continuous EMI Fields.
The amplitude of the secondary field at the receiver
is normalized to the amplitude of the primary field at the receiver, in units
such as percent, parts-per-thousand (ppt), etc. The phase of the secondary field is also compared to that of the
primary field.
Figure 3 is a portion of Figure 2, stretched to show
phase components of the secondary field.
The in-phase component coincides with the peak of the primary field, and
the quadrature component leads the in-phase component by one-quarter cycle,
i.e. 90 degrees.
Figure 3: Detail of Phase Components.
In this example, the secondary field leads the
primary field by about 45 degrees, so the in-phase and quadrature amplitudes
are about equal.
Transmitter-receiver geometry is an essential aspect
of EMI. Geometry is the orientation and
separation of the transmitter (Tx) and receiver (Rx) relative both to each
other and to the earth. DUALEM
instruments combine two geometries, usually called “horizontal co-planar” (HCP)
and “perpendicular” (PRP). The
following figure shows a schematic profile of a DUALEM instrument.
Figure 4: DUALEM Schematic
Profile.
A DUALEM instrument uses bobbin-wound coils for its
Tx and Rx. For the HCP geometry, the
windings of the Tx- and Rx-coils lie in the same horizontal plane. The PRP geometry shares the same Tx, but the
windings of its Rx coil are vertical, and axis of the coil intersects the Tx.
The Tx-Rx separation is 2 m for the DUALEM-2, and 4
m for the DUALEM-4. (As the separations
are more than several times as large as the coil diameters, each coil can be
treated mathematically as an oscillating magnetic dipole that is aligned with
the coil axis.)
To illustrate the significance of geometry, the
following figure shows schematic profiles at four critical points of a DUALEM
traverse over a thin-and-shallow conductor.
The senses of current in the conductor and in the DUALEM coils are shown
by the triangles on these features.
Triangles also show the senses of the magnetic flux of the primary and
secondary fields. At each point,
current in the edge of the Tx coil closest to us flows from right to left,
which generates a primary field with a downward sense.
At the bottom of the figure are diagrammatic
profiles that show the characteristics of continuous measurements across this
type of conductor.
Figure 5: Profiles of DUALEM and
Thin Conductor.
Profile 1 shows the DUALEM approaching the
conductor. The leftward sense of the
primary field that intersects the conductor induces current that, in the edge
of the conductor closest to us, flows upward.
This generates a secondary magnetic field with a similar sense through
the conductor to that of the primary field.
The secondary field disperses away from the axis of
the induced current, and some of its flux intersects the Rx coils. The downward sense of the flux induces
current in the HCP Rx coil that, in the edge closest to us, flows from right to
left. The leftward sense of the flux
induces current in the PRP Rx coil that, in the edge closest to us, flows
upward.
Relative to the Tx current, these Rx currents are
considered to be positive, but they are weak due to the combined distance that
the primary and secondary fields must travel, first to the conductor, and then
to the Rx. Their values are shown
qualitatively on the measurement profiles at the bottom of the figure, directly
below the location of the conductor in profile 1.
Profile 2 shows the point at which the axis of the
Tx is directly over the conductor. As
the flux of the primary field disperses symmetrically about the Tx axis, no net
amount of flux intersects the conductor, no current is induced, no secondary
field is generated, and no current is induced in the Rx coils. The Rx values of zero are plotted on the
measurement profiles, directly below the position of the conductor in profile
2.
Beyond the point of profile 2, the primary field has
a rightward sense. This sense induces
current in the conductor that, in the edge closest to us, flows downward.
Profile 3 shows the point at which the conductor is
midway between the Tx and Rx. The
downward sense of the conductor current induces a secondary field with a
rightward sense. Some of the secondary
flux disperses upward, develops an upward component to its sense, and
intersects the Rx.
The upward sense of the flux induces current in the
HCP Rx coil that, in the edge closest to us, flows from left to right. The rightward sense of the flux induces
current in the PRP Rx coil that, in the edge closest to us, flows
downward. The senses of these currents
are opposite those in profile 1 and, thus, are deemed negative.
As shown on the measurement profiles, the most
negative HCP value over a thin conductor generally occurs where the conductor
is midway between the Tx and Rx. This
is a consequence of the relatively short combined distance that the primary and
secondary fields must travel, but also to the relatively strong upward
component of the secondary flux at this point.
As the upward component increases, the rightward
component decreases, which decreases the amplitude of the PRP value. The secondary flux is vertical where the PRP
value passes through zero between the points of profiles 3 and 4.
Profile 4 shows the point at which the HCP Rx coil
is directly over the conductor. The
secondary field is horizontal above the conductor, so no net amount of flux
passes through the HCP Rx coil and the HCP current is zero.
For practical purposes in DUALEM instruments, the
PRP and HCP Rx coils are separated slightly.
At the point of profile 4, the PRP Rx coil has not moved directly over
the conductor, but the sense of the secondary flux through the coil is
leftward, producing a positive value for the current. The PRP response reaches a maximum near the point directly over
the conductor.
Beyond the point of profile 4, the HCP and PRP
values generally decrease, as the combined distance increases through which the
primary and secondary fields must travel.
Continuous responses of the types depicted here are
commonly termed “shoulders-and-trough” for HCP, and “crossover” for PRP. Note that the depictions are qualitative,
and that actual responses vary greatly in amplitude, phase, form and complexity
with, among other factors, the conductance, depth and shape of the conductor
and its situation relative to the survey path.
Lightning provides another graphic example of
geometry. Lightning generates broadband
electromagnetic energy in the atmosphere, which appears as noise in broadband
EMI data. (DUALEM instruments are
sensitive only to the small fraction of the energy that has an effective
frequency equal to the 9-kHz DUALEM frequency.)
If we characterize lightning as a vertical pulse of
electrical current, the current will transmit a horizontal magnetic field. Long-distance propagation of the field is
enabled where a conductive earth (e.g. ocean) is present to complement the
ionosphere in guiding the horizontal field.
The following figure is a plan view of a lightning strike, its
transmitted field, and the HCP and PRP geometries.
Figure 6: Plan of Geometry and
Atmospheric Noise.
The effect of the horizontal magnetic field on the
horizontal Tx and HCP Rx will be minimal, as little flux will intersect the
coils. With the long axis of the PRP
geometry oriented toward the lightning, as shown, little flux from the
horizontal magnetic field will intersect the PRP Rx so the effect, again, will
be minimal. Where the PRP geometry is
oriented at right angles to the prevalent direction of atmospheric disturbance,
however, noise in the PRP data may arise from distant disturbances if the
intervening earth is conductive, and noise typically increases as the distance
to the disturbance decreases.
Induction
Number
The induction number (IN) characterizes the EMI of a
homogeneous earth. The IN is defined as
(cmw)1/2s, where c is the conductivity of the earth, m is the
magnetic permeability of the earth, w is the angular frequency of the primary
field and s is the transmitter-receiver separation.
For example the IN is about 0.084, where an
instrument with 2-m separation operates at 9 kHz (2π x 9000 radians/s) on
an earth with 25 mS/m (0.025 S/m) conductivity and free-space (4π x 10-7
H/m) magnetic permeability. Note that
the IN is dimensionless.
The following figure shows the amplitudes of phase
components of the secondary field that are measured by the HCP and PRP
geometries at INs from 0 to 3.5. The
markers on the curves show increments of 0.5 in the IN. The chart is drawn from data in Frischknecht
(1967).
Figure 7: Phasor Diagram for HCP
and PRP.
The
phasor diagram provides a graphic summary of the effect of IN on the amplitude
and phase of the secondary field. Near
the origin, both the HCP and PRP curves show a steep positive slope, as
quadrature amplitude increases rapidly with increasing, but low, induction.
As induction continues to increase, both components
increase in amplitude, but the slopes of the curves start to flatten as the
peak of secondary field begins to shift towards the in-phase component.
At INs of about 0.76 for HCP and 1.8 for PRP, the
curves are horizontal. At these INs,
the increase in quadrature due to increasing induction is balanced by the
decrease in quadrature due to the shift of the secondary-field peak toward
in-phase.
At INs of about 1.2 for HCP and 3.3 for PRP,
quadrature has decreased to zero, as the secondary field has shifted to exact
alignment with the in-phase component.
Also at an IN of about 3.3 for PRP, and about 1.7
for HCP, the curves become vertical.
The secondary field has shifted beyond in-phase alignment, and the
decrease in in-phase amplitude due to the shift balances the increase due to
increasing induction.
The HCP curve passes through another notable point
before reaching the maximum graphed IN of 3.5.
At an IN of about 2.6, HCP in-phase is zero, as the secondary field has
shifted to an alignment exactly opposite to quadrature.
The accurate measurement of phase components over a
significant range of INs provides a basis for interpreting the extent of
conductive layering in the earth. The
following figure shows an example of such interpretation by Palacky (1991).
Figure 8: Multi-frequency HCP
Sounding.
The
figure shows again the curve for HCP measurements on a homogeneous earth for
INs of 0 to 3.5, along with eight pairs of phase measurements (“100 m”). The
measurements were made at the eight frequencies of 110-, 220-, 440-, 880-,
1760-, 3520-, 7040-, and 14080-Hz, using an HCP array with 100-m separation
between the Tx and Rx. The separation
was controlled within a tolerance of 0.2 m, which constrained the error of
repeated measurements to about 10 ppt.
Frequent calibrations were used to correct for drift.
The eight measurements conform fairly well to the
curve for a homogenous earth. The
measurement at 7040 Hz lies very close to the point of the curve with an IN of
about 2.2. Using the formula IN = (cmw)1/2s,
the measurement indicates a conductivity
of about 9 mS/m for a homogenous earth.
Conductive layering in the earth is revealed by the
pattern of the small deviations of the measurements from the curve. The measurement at 14080 Hz lies farther
from the origin than the local part of the curve, while the 3 measurements below
7040 Hz, i.e. at 3520-, 1760- and 880-Hz, lie closer to the origin than the
curve. (The measurements at 440 Hz and
below are poorly resolved near the origin, and are not useful for
interpretation.)
The deviations from the curve indicate that the
earth seems more conductive than a homogeneous earth at the
shallower-penetrating high frequency, while the earth seems less conductive at
the deeper-penetrating lower frequencies.
Thus, the deviations suggest that the upper earth is more conductive
than the lower earth.
When the measurements are compared to model values
for an earth with a surficial layer of 20-mS/m conductivity and 42-m thickness
on underlying material of 0.1-mS/m conductivity, the measurements fit with an
error of about 1 %. Such an earth
corresponds well with a borehole at the measurement location, which reached
crystalline bedrock beneath 35 m of glacial sediments.
Large Tx-Rx separations are a means of obtaining
measurements at the range of INs necessary for multi-frequency sounding. However, spatial resolution coarsens as separation
increases, which makes large separations impractical for sounding the uppermost
10 m of the earth.
Theoretically, it should be possible to sound
shallowly with phase measurements from short Tx-Rx separations, provided that
the measurement frequencies cover a substantial range of INs. Inspection of the IN formula, however, shows
that this is generally impractical. For
example, if we shorten the separation by a factor of 50, e.g. from 100 m to 2
m, we must broaden the frequency range by a factor of 2500, as IN is
proportional to separation, but to only the square-root of frequency.
The following figure shows an attempt at
multi-frequency sounding with an HCP instrument that has a Tx-Rx separation of
1.6 m. The figure shows the three pairs
of phase measurements (“1.6 m”) made at 990-, 4470-, and 19950-Hz. The figure also shows the HCP curve for a
homogeneous earth, with the IN range of 0 to 0.05.
Figure 9: Multi-frequency at
Short Separation.
The scatter of the measurements arises from in-phase
values that are unrealistic in sign and amplitude. The values indicate that errors in zero-level calibration are
much larger than the in-phase responses of the earth, and that the errors vary
incoherently from frequency to frequency.
In-phase error renders academic the question of the
IN range covered by the frequencies.
(For a range comparable to the previous example, the 1.6-m instrument
would have to operate at frequencies up to 98 MHz, a frequency at which
ground-penetrating-radar instruments operate using EM radiation, rather than
EMI). More significantly, the scatter
of the measurements causes the simple interpretation of homogeneous-earth
conductivity from the multi-frequency data to become problematic.
EMI instruments with short Tx-Rx separations are
generally incapable of measuring in-phase with interpretable accuracy, as the
strength of the primary field at the Rx is inversely proportion to the cube of
the separation. For example, the
primary field at the Rx of a 1.6-m array is 244,000-times as strong as it is at
the Rx of a 100-m array, and in-phase zero-level instability caused by
expansion, contraction or bending of the array is amplified accordingly.
Less importantly, fluctuations in the magnetic
susceptibility of shallow material will affect the stability of the in-phase
component.
Low Induction
Number
At LIN, Wait (1962) shows that nearly all of the
conductive response from a homogeneous- or layered-earth is quadrature, so an
inaccurate in-phase component can be ignored.
Wait proposes that LIN conditions exist for the PRP geometry to an IN of
about 0.5, where quadrature still accounts for about 99 % of total response. Using a similar criterion for the HCP
geometry, LIN conditions exist to an IN of about 0.16.
Using these limits and assuming that the earth has
the magnetic permeability of free space, LIN conditions exist for DUALEM-2
measurements where HCP conductivity does not exceed 90 mS/m, and PRP
conductivity does not exceed 800 mS/m.
The corresponding conductivities for the DUALEM-4 are 23 mS/m for HCP,
and 210 mS/m for PRP.
Wait (ibid.) shows that conductivity at LIN is
linearly proportional to quadrature, and describes the accumulation of PRP
conductivity from a layered earth.
McNeill (1980) describes the corresponding accumulation of HCP
conductivity. These accumulations form
the basis of interpreting conductive layering in the upper few metres of the
earth from DUALEM measurements.
The PRP geometry accumulates its measurement of
conductivity with depth proportionally to P(d)=2d/(4d2+1)1/2,
where d is the depth beneath
the instrument, in units of the Tx-Rx separation.
In the same units of Tx-Rx separation, the HCP
geometry accumulates its measurement of conductivity proportionally to
H(d)=1-1/(4d2+1)1/2.
The accumulation functions provide an indication of
depth of exploration. For example, the
PRP geometry accumulates 70 % of its total response within a depth equivalent
to 0.5 coil-separations, and the HCP geometry accumulates similar response
within 1.5 coil-separations.
Where there is conductive layering beneath the
instrument, each geometry will measure a value of conductivity equal to the sum
of the conductivity of each layer times the accumulation of response in that
layer.
The following figure depicts a layered earth sounded
by a DUALEM-4, where h is the height of the instrument above the earth, ca
is the conductivity of the air, t is the thickness of an upper layer of the
earth, ct is the conductivity of the upper layer, and c is the
conductivity of the remainder of the earth.
Figure 10: Sounding a Layered
Earth.
The PRP conductivity measured over this earth will
be:
P = ca P(h) + ct (P(h+t) - P(h))
+ c (P(∞) - P(h+t))
and the HCP conductivity will be:
H = ca H(h) + ct (H(h+t) -
H(h)) + c (H(∞) - H(h+t)).
Since the conductivity of air is essentially zero,
and the accumulation of response to infinity is 1, the expressions for apparent
conductivity simplify to:
P = ct (P(h+t) - P(h)) + c (1 - P(h+t))
and
H = ct (H(h+t) - H(h)) + c (1 - H(h+t)).
If we know the height of measurement, and assign a
thickness for the upper portion of the earth, we can calculate the various
accumulations of response. This leaves
us with two unknowns, ct and c, in two equations, which can be
solved by the isolation and substitution of variables.
If we assign a value for the conductivity of either
the upper- or the underlying-earth, we should be able to solve for the other
conductivity and the thickness of the upper earth. However, the variable t is embedded in the accumulation functions
in such a way that it is difficult to isolate or substitute. Nevertheless, where one of the
conductivities can be fixed at a realistic value, approximate solutions for
thickness and the other conductivity are sometimes satisfactory.
To increase confidence in an approximate solution,
or to estimate values for more than two variables, measurements can be made at
a number of heights. Such a procedure
performed at a given location is called a vertical sounding.
The following figure shows DUALEM-2 measurements
(HCP and PRP) from a vertical sounding, along with lines that show values
calculated from the accumulation functions for both a homogeneous earth (HCP HE
and PRP HE) and a layer on an underlying earth (HCP L+E and PRP L+E).
Figure 11: Vertical Sounding and
Models.
The HCP and PRP conductivities decrease with
increasing height, as more of the responses accumulate in non-conductive
air. The dashed lines show the values
that should be obtained over this range of heights above a 32-mS/m homogenous
earth.
The HCP geometry accumulates more of its response at
greater depth than the PRP geometry. As
the HCP measurements are greater than those for a 32-mS/m earth, and the PRP
measurements are less, the sounding indicates the presence of a lower earth
that is more conductive than 32 mS/m, under a less conductive surficial layer.
The solid lines show the values that should be
obtained over a model earth with such layering. The model has a surficial layer of 1-mS/m conductivity and 0.55-m
thickness, on an underlying earth of 44-mS/m conductivity.
While the presence of a surficial layer is indicated
clearly by the vertical sounding, the parameters of the layer are poorly
resolved. For example, a model with a
layer of 5-mS/m conductivity and 0.65-m thickness on an underlying earth of
45-mS/m conductivity fits the measurements essentially as well as illustrated
example.
The vertical sounding was taken over an earth with
turf and topsoil on fine-grained subsoil that becomes rich in carbonate below a
horizon of leaching.
The vertical sounding was made at the same location
as the multi-frequency measurements of figure 9. The conductivities of the sounding confirm that the
multi-frequency measurements were made at LIN, regardless of frequency.
At LIN, in-phase amplitudes are an insignificant
fraction of total response. Since the
in-phase values measured by the multi-frequency instrument are similar in
amplitude to the quadrature values, they are spurious and useless in the
interpretation of conductivity.
Indeed, it is noteworthy that short multi-frequency
instruments that show measurements of apparent conductivity do so by scaling
the values from the amplitude of the quadrature component, an interpretation
that is valid only at LIN. It follows
that if these measurements are valid, they must accumulate from depths defined
by the LIN functions of Tx-Rx separation, regardless of the operating frequency
of the instrument.
Summary
Many geological materials, including most soils and
some bedrock, have conductivities that can be measured with EMI. Continuous
EMI, which measures in-phase and quadrature responses with various Tx-Rx
geometries, is an efficient technique both for profiling discrete conductors
and for sounding conductivity.
Where Tx-Rx separation is large, measurements over a
wide range of frequencies can be used to estimate the thickness and
conductivity of contrasting layers in the earth. Where Tx-Rx separation is small, layer thickness and conductivity
can be estimated from quadrature measurements using distinct geometries and
measurement heights.
References
Frischknecht, F.C., 1967, Fields about an
oscillating magnetic dipole over a two-layer earth, and application to ground
and airborne electromagnetic surveys: Colorado School of Mines Quarterly, 62.
McNeill, J.D., 1980, Electromagnetic terrain
conductivity measurement at low induction numbers: Geonics Ltd., Technical Note
TN-6.
Palacky, G.J., 1987, Resistivity characteristics of
geologic targets, in Nabighian, Misac N., ed., Electromagnetic Methods
in Applied Geophysics (2 volumes): Society of Exploration Geophysicists, I, 53-129.
Palacky, G.J., 1991, Application of the
multifrequency horizontal-loop EM method in overburden investigations,
Geophysical Prospecting, 39, 1061-82.
Wait, J.R., 1962, A note on the electromagnetic
response of a stratified earth: Geophysics, 27, 382-85.